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Mirrors > Home > MPE Home > Th. List > domnnzr | Structured version Visualization version GIF version |
Description: A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.) |
Ref | Expression |
---|---|
domnnzr | ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2610 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | eqid 2610 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
4 | 1, 2, 3 | isdomn 19115 | . 2 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = (0g‘𝑅) → (𝑥 = (0g‘𝑅) ∨ 𝑦 = (0g‘𝑅))))) |
5 | 4 | simplbi 475 | 1 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 .rcmulr 15769 0gc0g 15923 NzRingcnzr 19078 Domncdomn 19101 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-domn 19105 |
This theorem is referenced by: domnring 19117 opprdomn 19122 abvn0b 19123 fidomndrng 19128 domnchr 19699 znidomb 19729 nrgdomn 22285 ply1domn 23687 fta1glem1 23729 fta1glem2 23730 fta1b 23733 lgsqrlem4 24874 idomrootle 36792 deg1mhm 36804 |
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